All of the 4th grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$8.00$ each for teachers and $$3.50$ each for students, and the group paid $$40.50$ in total. The next month, the same group visited a science museum where the tickets cost $$16.00$ each for teachers and $$10.50$ each for students, and the group paid $$105.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+3.5y = 40.5}$ ${16x+10.5y = 105.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-16x-7y = -81}$ ${16x+10.5y = 105.5}$ Add the top and bottom equations together. $ 3.5y = 24.5 $ $ y = \dfrac{24.5}{3.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {8x+3.5y = 40.5}$ to find $x$ ${8x + 3.5}{(7)}{= 40.5}$ $8x+24.5 = 40.5$ $8x = 16$ $x = \dfrac{16}{8}$ ${x = 2}$ You can also plug ${y = 7}$ into $ {16x+10.5y = 105.5}$ and get the same answer for $x$ ${16x + 10.5}{(7)}{= 105.5}$ ${x = 2}$ There were $2$ teachers and $7$ students on the field trips.